ENSAE Paris - École d’ingénieurs pour l’économie, la data science, la finance et l’actuariat

Méthodes numériques en ingénierie financière

Objectif

Participants of this course will master computational techniques frequently used in mathematical finance applications.

 

Prerequisites: Stochastic processes / stochastic calculus, Numerical Analysis, Derivatives, Command of Python

TDs: In the form of ipython notebooks

Plan

1. Brief introduction to asset price modeling and option pricing

  • Basic stochastic models in finance
  • Introduction to numerical methods for option pricing

2.  Monte Carlo methods for pricing

  • Euler schemes and applications to volatility models
  • Variance reduction 

3. Option pricing via PDEs

  • Finite difference approximation of Black-Scholes PDE
  • American options and free boundary problems

4. Transformation based methods

  • Affine models
  • Option pricing via Fourier transforms

5. Density approximation techniques

  • Polynomial models and calculation of moments
  • Option pricing via density approximation

6. Rough Volatility

  • Motivation for rough volatility models and examples
  • The rough Heston model 

7. Machine learning methods for pricing

  • Parametric option pricing and calibration with Neural Networks
  • Neural networks for solving PDEs

Références

·      Achdou, Yves; Pironneau, Olivier. Computational methods for option pricing. Frontiers in Applied Mathematics, 30. SIAM, Philadelphia, PA, 2005.

·      Björk, Tomas. Arbitrage theory in continuous time. Third edition, OUP Oxford, 2009.

·      Gatheral,  Jim. Volatility Surface: A Practitioner’s Guide. Wiley, 2006.

·      Glasserman, Paul. Monte Carlo methods in financial engineering. Springer, 2003.

·      Hilber, Norbert; Reichmann, Oleg; Schwab, Christoph; Winter, Christoph. Computational methods for quantitative finance. Springer, 2013.

·      Hirsa, Ali. Computational methods in finance. Chapman & Hall/CRC Financial Mathematics Series. CRC Press, Boca Raton, FL, 2013.

·      Lamberton, Damien; Lapeyre, Bernard. Introduction to stochastic calculus applied to finance. Second revised edition. Chapman & Hall/CRC, 2008.

·      Seydel, Rüdiger U. Tools for computational finance. Fourth edition. Universitext. Springer-Verlag, Berlin, 2009.

·      Shreve, Steven E. Stochastic calculus for finance II: Continuous-Time models, Volume 11. Springer Science & Business Media, 2004.

·      Additional lecture material will be provided by the instructors.